A218300 E.g.f. A(x) satisfies: A( x/(exp(x)*cosh(x)) ) = exp(2*x)*cosh(2*x).

1, 2, 12, 104, 1216, 18112, 329600, 7108096, 177549312, 5046554624, 160947232768, 5694342479872, 221410157133824, 9387011838312448, 431051678297358336, 21316106766591721472, 1129526392342026649600, 63855305138514241257472, 3836490516381680506241024

Offset

0,2

Comments

More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),

then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Formula

E.g.f.: A(x) = Sum_{n>=0} 2*(n+2)^(n-1) * cosh((n+2)*x) * x^n/n!.

E.g.f.: A(x) = 1 + Sum_{n>=0} 2*(n+2)^(n-1) * sinh((n+2)*x) * x^n/n!.

a(n) ~ c * n^(n-1) / (exp(n) * (LambertW(exp(-1)))^n), where c = sqrt(1 + LambertW(exp(-1)))/LambertW(exp(-1))^2 = 14.5815783688217906961670551786416446... . - Vaclav Kotesovec, Jul 13 2014, updated Jun 10 2019

Example

E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 104*x^3/3! + 1216*x^4/4! + 18112*x^5/5! +...
where
A(x) = cosh(2*x) + 2*3^0*cosh(3*x)*x + 2*4^1*cosh(4*x)*x^2/2! + 2*5^2*cosh(5*x)*x^3/3! + 2*6^3*cosh(6*x)*x^4/4! + 2*7^4*cosh(7*x)*x^5/5! +...

Mathematica

nmin = 0; nmax = 18; sol = {a[0] -> 1}; nsol = Length[sol];
Do[A[x_] = Sum[a[k] x^k/k!, {k, 0, n}] /. sol; eq = CoefficientList[ A[x/(Exp[x] Cosh[x])] - Exp[2x] Cosh[2x] + O[x]^(n+1), x][[nsol+1;; ]] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, nsol+1, nmax}];
a /@ Range[nmin, nmax] /. sol (* Jean-François Alcover, Nov 06 2019 *)

Prog

(PARI) {a(n)=local(Egf=1, X=x+x*O(x^n), R=serreverse(x/(exp(X)*cosh(X)))); Egf=exp(2*R)*cosh(2*R); n!*polcoeff(Egf, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* Formula derived from a LambertW identity: */
{a(n)=local(Egf=1, X=x+x*O(x^n)); Egf=sum(k=0, n, 2*(k+2)^(k-1)*cosh((k+2)*X)*x^k/k!); n!*polcoeff(Egf, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A201595, A218301, A218302, A218303, A218304, A218305, A218306, A218307, A218308, A218309, A218310, A217900.

Cf. A202357.

Sequence in context: A194951 A104533 A216794 * A125031 A258230 A085867

Adjacent sequences: A218297 A218298 A218299 * A218301 A218302 A218303

Keyword

nonn

Author

Paul D. Hanna, Oct 25 2012

Status

approved